| Tschirnhaus-Weierstrass curves - Josef Schicho We define the concept of Tschirnhaus-Weierstrass curve, named after the Weierstrass form of an elliptic curve and Tschirnhaus transformations. Every pointed curve has a Tschirnhaus-Weierstrass form, and this representation is unique up to a scaling of variables. This is useful for computing isomorphisms between curves. Downloads: 2 | |

| Transitivity on Weierstrass points - Zoe Laing We look for Riemann surfaces whose automorphism group acts transitively on the Weierstrass points. We concentrate on hyperelliptic surfaces, surfaces with PSL(2, q) as automorphism group, Platonic surfaces and Fermat curves. Downloads: 2 | |

| Stone-Weierstrass Theorem - Guy Laville It will be shown that the Stone-Weierstrass theorem for Clifford-valued functions is true for the case of even dimension. It remains valid for the odd dimension if we add a stability condition by principal automorphism. Downloads: 3 | |

| Weierstrass's Non-Differentiable Function (Volume 17) - Hardy, G. H. "Weierstrass's Non-Differentiable Function" is an article from Transactions of the American Mathematical Society, Volume 17. View more articles from Transactions of the American Mathematical Society.View this article on JSTOR.View this article's JSTOR metadata.You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1989005 Downloads: 40 | |

| Weierstrass's Elliptic Integral (Volume 6) - Fiske, Thomas S. "Weierstrass's Elliptic Integral" is an article from The Annals of Mathematics, Volume 6. View more articles from The Annals of Mathematics.View this article on JSTOR.View this article's JSTOR metadata.You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967469 Downloads: 22 | |

| Relativistic Weierstrass random walks - Alberto Saa The Weierstrass random walk is a paradigmatic Markov chain giving rise to a L\'evy-type superdiffusive behavior. It is well known that Special Relativity prevents the arbitrarily high velocities necessary to establish a superdiffusive behavior in any process occurring in Minkowski spacetime, implying, in particular, that any relativistic Markov chain describing spacetime phenomena must be essentially Gaussian... Downloads: 4 | |

| Weierstrass P-lumps - RJ Cova We present our results of a numerical investigation of the behaviour of a system of two solitons in the (2+1) dimensional $CP^1$ model on a torus. Defined by the elliptic function of Weierstrass, and working in the Skyrme version of the model, the soliton lumps exhibit splitting, scattering at right angles and motion reversal in the various configurations considered. The work is restricted to systems with no initial velocity. Downloads: 3 | |

| The Weierstrass Substitution in Integration - Patrick Bruskiewich To simplify an integral that is a rational function in cos(x) or sin(x), a substitution of the form t = tan(ax/2) will convert the integrand into an ordinary rational function in t. This substitution, is known as the Weierstrass Substitution, and honours the mathematician, Karl Weierstrass (1815-1897) who developed the technique. Keywords: Weierstrass Substitution; Integration; Calculus; mathematics; mathematician; cosine; sine; rational expression; transformation Downloads: 17 | |

| Weierstrass integrability of differential equations - Jaume Giné The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define {\it Weierstrass integrability} and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non--Liouvillian integrable systems. Downloads: 1 | |

| The Proportion of Weierstrass Semigroups - Nathan Kaplan We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the family of semigroups known to be Weierstrass semigroups using a result of Eisenbud and Harris, has zero density in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup. Downloads: 4 | |

| Weierstrass preparation and algebraic invariants - David Harbater We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period-index problem for division algebras, over fraction fields of complete two-dimensional rings... Downloads: 2 | |

| The Significance of Weierstrass's Theorem (Volume 20) - Hedrick, E. R. "The Significance of Weierstrass's Theorem" is an article from The American Mathematical Monthly, Volume 20. View more articles from The American Mathematical Monthly.View this article on JSTOR.View this article's JSTOR metadata.You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2974105 Downloads: 25 | |

| On the Stone-Weierstrass theorem - Silviu Teleman This paper extends a version of the Stone-Weierstrass theorem to more general C*-algebras. Namely, assume that A is a unital, not necessarily separable, C*-algebra, and B is a C*-subalgebra containing the unit element. Then, I prove that: If B separates the factorial states of A, then B=A. This generalizes a result of Popa and Longo for the case when A is separable. A true Stone-Weierstrass theorem would state that, if B separates the pure states of A, then B=A... Downloads: 11 | |

| Limits of special Weierstrass points - Caterina Cumino Let C be the union of two general connected, smooth, nonrational curves X and Y intersecting transversally at a point P. Assume that P is a general point of X or of Y. Our main result, in a simplified way, says: Let Q be a point of X. Then Q is the limit of special Weierstrass points on a family of smooth curves degenerating to C if and only if Q is not P and either of the following conditions hold: Q is a special ramification point of the linear system |K_X+(g_Y+1)P|, or Q is a ramification poi... Downloads: 1 | |

| On maximal curves having classical Weierstrass gaps - Arnaldo Garcia We study geometrical properties of maximal curves having classical Weierstrass gaps. Downloads: 1 | |

| On Weierstrass points and optimal curves - Rainer Fuhrmann We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves. Downloads: 4 | |

| On the constellations of Weierstrass points - Fernando Torres We prove that the constellation of Weierstrass points characterizes the isomorphism-class of double covering of curves of genus large enough. Downloads: 6 | |

| Hodge structures and Weierstrass $σ$-function - Grzegorz Banaszak In this paper we introduce new definition of Hodge structures and show that $\R$-Hodge structures are determined by $\R$-linear operators that are annihilated by the Weierstrass $\sigma$-function | |

| On Relations of Hyperelliptic Weierstrass al Functions - Shigeki Matsutani We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$ as solutions of sine-Gordon equation using Weierstrass's local parameters, which are characterized by two ramified points. Though the hyperelliptic solutions of the sine-Gordon equation had already obtained, our derivations of them are simple; they need only residual computations over the curve and primitive matrix computations. Downloads: 3 | |

| A note on the generalized Weierstrass representation - L. Martina The study of the relation between the Weierstrass inducing formulae for constant mean curvature surfaces and the completely integrable euclidean nonlinear sigma-model suggests a connection among integrable sigma -models in a background and other type of surfaces. We show how a generalization of the Weierstrass representation can be achieved and we establish a connection with the Weingarten surfaces... Downloads: 3 | |

| On functions of Jacobi and Weierstrass (I) - Yu. V. Brezhnev We present some new results in theory of classical theta-functions of Jacobi and sigma-functions of Weierstrass: ordinary differential equations (dynamical systems) and series expansions. The paper is basically organized as a stream of new formulas. | |

| Two remarks on the Weierstrass flag - Enrico Arbarello We show that the locally closed strata of the Weierstrass flags on the moduli spaces of curves of genus g and on the moduli space of curves of genus g with one marked point are almost never affine. Downloads: 5 | |

| The Weierstrass representation for pluriminimal submanifolds - C. Arezzo In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorphic manner. Downloads: 2 | |

| Local invariants attached to Weierstrass points - Robin de Jong Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation of the hyperelliptic discriminant of X/S, and the valuation of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable these invariants are known to satisfy certain inequalities... Downloads: 2 | |

| Multivortex Solutions of the Weierstrass Representation - P. Bracken The connection between the complex Sine and Sinh-Gordon equations on the complex plane associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlev\'e test and analyse the possibility of deriving the B\"acklund transformation from the singularity analysis of the complex Sine-Gordon equation. We make use of the analysis using the known relations for the Painlev\'{e} equations to construct e... Downloads: 2 | |

| Mixed Hodge structures and Weierstrass $σ$-function - Grzegorz Banaszak A $\sigma$-operator on a complexification $V_{\C}$ of an $\R$-vector space $V_{\R}$ is an operator $A \in \rm{End}_{\C} (V_{\C})$ such that $\sigma (A) = 0$ where $\sigma (z)$ denotes the Weierstrass $\sigma$-function. In this paper we define the notion of the strongly pseudo-real $\sigma$-operator and prove that there is one to one correspondence between real mixed Hodge structures and strongly pseudo-real $\sigma$-operators. | |

| Generalized Weierstrass Relations and Frobenius reciprocity - Shigeki Matsutani This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dirac operator, gives the data of the immersion. In the derivation, the simple Frobenius reciprocity of Clifford algebras $S$ and $M$ plays important roles. | |

| Gaussian curvature at the hyperelliptic Weierstrass points - Abel Castorena Let C be a hyperelliptic Riemann surface. We show that the hyperelliptic Weierstrass points of C are non-degenerated critical points of Morse index +2 of the curvature function K of the Theta metric on C (called also Bergman metric). When the genus of C is two, we compute all critical points of K and we show in this case that K is a Morse function. Downloads: 1 | |

| A survey on the Weierstrass approximation theorem - Dilcia Perez The celebrated and famous Weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. This theorem is the first significant result in Approximation Theory of one real variable and plays a key role in the development of General Approximation Theory. Our aim is to investigate some new results relative to such theorem, to present a history of the subject, and to introduce some open problems. Downloads: 9 | |

| Automorphisms of Curves and Weierstrass semigroups - Sotiris Karanikolopoulos The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group $G$ we introduce a new filtration of the group decomposition subgroup of $G$. The relation to the ramification filtration is investigated in the case of cyclic covers. We relate our results to invariants defined by Boseck and we study the one point ramification case. We also give applications to Hasse-Witt invariant and symmetric semigroups. Downloads: 18 | |

| Weierstrass's criterion and compact solitary waves - Michel Destrade Weierstrass's theory is a standard qualitative tool for single degree of freedom equations, used in classical mechanics and in many textbooks. In this Brief Report we show how a simple generalization of this tool makes it possible to identify some differential equations for which compact and even semicompact traveling solitary waves exist. In the framework of continuum mechanics, these differential equations correspond to bulk shear waves for a special class of constitutive laws. Downloads: 12 | |

| Analysis I- The Weierstrass Pathological Function - Katrin Wehrheim All known examples of non-differentiable continuous functions are constructed in a similar fashion to the following example � they are limits of functions that oscillate more and more on small scales, but with higher-frequency oscillations being damped quickly. The example we give here is a faithful reproduction of Weierstrass�s original 1872 proof. It is somewhat more complicated than the example given as Theorem 7.18 in Rudin, but is superior in at least one important way, as explained in ... Keywords: Maths; Analysis and Calculus; Mathematics Downloads: 5 | |

| Homogeneous division polynomials for Weierstrass elliptic curves - Jinbi Jin Starting from the classical division polynomials we construct homogeneous polynomials a_n, b_n, c_n such that for P = (x:y:z) on an elliptic curve in Weierstrass form over an arbitrary ring we have nP = (a_n(P):b_n(P):c_n(P)). To show that a_n, b_n, c_n indeed have this property we use the a priori existence of such polynomials, which we deduce from the Theorem of the Cube. Downloads: 4 | |

| Analysis I- The Weierstrass Pathological Function - Katrin Wehrheim Until Weierstrass published his shocking paper in 1872, most of the mathematical world (including luminaries like Gauss) believed that a continuous function could only fail to be differentiable at some collection of isolated points. In fact, it turns out that �most� continuous functions are non-differentiable at all points. (To understand what this statement could mean, you should take courses in topology and measure theory.) However, Weierstrass was not, in fact, the �rst to construct suc... Keywords: Maths; Analysis and Calculus; Mathematics Downloads: 11 | |

| Weierstrass solutions for dissipative BBM equation - Stefan C. Mancas In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified BBM equation. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant make the equation integrable in terms of Weierstra... Downloads: 8 | |

| A Weierstrass-type theorem for homogeneous polynomials - David Benko By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R^d. In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like origin-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approx... Downloads: 1 | |

| On higher genus Weierstrass sigma-function - Dmitry Korotkin The goal of this paper is to propose a new way to generalize the Weierstrass sigma-function to higher genus Riemann surfaces. Our definition of the odd higher genus sigma-function is based on a generalization of the classical representation of the elliptic sigma-function via Jacobi theta-function. Namely, the odd higher genus sigma-function $\sigma_{\chi}(u)$ (for $u\in \C^g$) is defined as a product of the theta-function with odd half-integer characteristic $\beta^{\chi}$, associated with a spi... | |

| Seminar in Geometry- Chapter 18- Weierstrass-Enneper Representations - Emma Carberry Weierstrass-Enneper Representations of Minimal Surfaces - Let M be a minimal surface de�ned by an isothermal parameterization... Keywords: Maths; Linear Algebra and Geometry; Geometry; Mathematics Downloads: 12 | |

| Weierstrass cycles in moduli spaces and the Krichever map - Jia-Ming We analyze cohomological properties of the Krichever map and use the results to study Weierstrass cycles in moduli spaces and the tautological ring. Downloads: 3 | |

| Representations of cyclic groups in positive characteristic and Weierstrass semigroups - Sotiris Karanikolopoulos We study the $k[G]$-module structure of the space of holomorphic differentials of a curve defined over an algebraically closed field of positive characteristic, for a cyclic group $G$ of order $p^\ell n$. We also study the relation to the Weierstrass semigroup for the case of Galois Weierstrass points. Downloads: 18 | |

| "Algebraic truths" vs "geometric fantasies": Weierstrass' Response to Riemann - Umberto Bottazzini In the 1850s Weierstrass succeeded in solving the Jacobi inversion problem for the hyper-elliptic case, and claimed he was able to solve the general problem. At about the same time Riemann successfully applied the geometric methods that he set up in his thesis (1851) to the study of Abelian integrals, and the solution of Jacobi inversion problem. In response to Riemann's achievements, by the early 1860s Weierstrass began to build the theory of analytic functions in a systematic way on arithmetic... Downloads: 1 | |

| Corrugated surfaces with slow modulation and quasiclassical Weierstrass representation - B. G. Konopelchenko Quasiclassical generalized Weierstrass representation for highly corrugated surfaces with slow modulation in the three-dimensional space is proposed. Integrable deformations of such surfaces are described by the dispersionless Veselov-Novikov hierarchy. Downloads: 8 | |

| The Weierstrass representation of closed surfaces in $R^3$ - Iskander A. Taimanov In the present paper which a sequel to dg-ga/9511005 and dg-ga//9610013 a global Weierstrass representation of an arbitrary closed oriented surface of genus $\geq 1$ in the the three-space is constructed. The Weierstrass spectrum of a torus immersed into $R^3$ is introduced and finite-zone planes as well as finite-zone solutions to the modified Novikov-Veselov equations are constructed. Downloads: 1 | |

| The Weierstrass subgroup of a curve has maximal rank - Martine Girard We show that the Weierstrass points of the generic curve of genus $g$ over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian. | |

| Generalized Weierstrass representation for surfaces in multidimensional Riemann spaces - B. G. Konopelchenko Generalizations of the Weierstrass formulae to generic surface immersed into $R^4$, $S^4$ and into multidimensional Riemann spaces are proposed. Integrable deformations of surfaces in these spaces via the modified Veselov-Novikov equation are discussed. Downloads: 1 | |

| Weierstrass representations for harmonic morphisms on Euclidean spaces and spheres - P. Baird We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic morphisms from Euclidean spaces and spheres. Downloads: 7 | |

| Note on the Weierstrass Preparation Theorem in Quasianalytic Local Rings - Jean-Philippe Rolin Consider quasianalytic local rings of germs of smooth functions closed under composition, implicit equation, and monomial division. We show that if the Weierstrass Preparation Theorem holds in such a ring then all elements of it are germs of analytic functions. . Downloads: 5 | |

| Application of Weierstrass units to relative power integral bases - Ho Yun Jung Let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. We construct relative power integral bases between certain abelian extensions of $K$ in terms of Weierstrass units. Downloads: 5 | |

| The Application of Weierstrass elliptic functions to Schwarzschild Null Geodesics - G. W. Gibbons In this paper we focus on analytical calculations involving null geodesics in some spherically symmetric spacetimes. We use Weierstrass elliptic functions to fully describe null geodesics in Schwarzschild spacetime and to derive analytical formulae connecting the values of radial distance at different points along the geodesic. We then study the properties of light triangles in Schwarzschild spacetime and give the expansion of the deflection angle to the second order in both $M/r_0$ and $M/b$ wh... Downloads: 2 | |

| Generalized Weierstrass Kernels on the Intersection of Two Complex Hypersurfaces - Franco Ferrari On plane algebraic curves the so-called Weierstrass kernel plays the same role of the Cauchy kernel on the complex plane. A straightforward prescription to construct the Weierstrass kernel is known since one century. How can it be extended to the case of more general curves obtained from the intersection of hypersurfaces in a $n$ dimensional complex space? This problem is solved in this work in the case $n=3$... | |